Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation

2019 ◽  
Vol 242 (5) ◽  
pp. 595-607 ◽  
Author(s):  
Yu. Yu. Bagderina
Keyword(s):  
Necessary Conditions ◽  
Second Order ◽  
Painlevé Equation ◽  
Painleve Equation

Hyperasymptotics for nonlinear ODEs II. The first Painlevé equation and a second-order Riccati equation

2005 ◽  
Vol 461 (2062) ◽  
pp. 3005-3021 ◽  
Author(s):  
A.B Olde Daalhuis
Keyword(s):  
Differential Equations ◽  
Riccati Equation ◽  
Second Order ◽  
Painlevé Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
Positive Real ◽  
Nonlinear Odes ◽  
First Order ◽  
Sheet Structure ◽  
Painleve Equation

This paper is a sequel to Olde Daalhuis (Olde Daalhuis 2005 Proc. R. Soc. A 461 , 2503–2520) in which we constructed hyperasymptotic expansions for a simple first-order Riccati equation. In this paper we illustrate that the method also works for more complicated nonlinear ordinary differential equations, and that in those cases the Riemann sheet structure of the so-called Borel transform is much more interesting. The two examples are the first Painlevé equation and a second-order Riccati equation. The main tools that we need are transseries expansions and Stokes multipliers. Hyperasymptotic expansions determine the solutions uniquely. Some details are given about solutions that are real-valued on the positive real axis.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

scholarly journals On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2095
Author(s):  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Jordan Algebra ◽  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Jordan Product ◽  
Completely Integrable ◽  
Matrix Generalization ◽  
Backlund Transformations ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Multiple Field

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.

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